GCD of Integer and its Negative

Theorem

Let $a \in \Z$ be an integer.

Then:

$\gcd \set {a, -a} = \size a$

where:

$\gcd$ denotes greatest common divisor

$\size a$ denotes the absolute value of $a$.

Proof

From Integer Divisor Results, the divisors of $a$ include $a$ itself.

From Integer Divides its Negative, $a \divides \paren {-a}$.

Thus we have:

$a \divides a$

and:

$a \divides -a$

and so:

$\gcd \set {a, -a} \ge \size a$

From Absolute Value of Integer is not less than Divisors, there is no divisor of $a$ which is greater than $a$.

That is:

$\gcd \set {a, -a} \le \size a$

Hence the result.

$\blacksquare$